Problems
For three sets \(A,B,C\) prove that \(A - (B\cap C) = (A-B)\cup (A-C)\). Draw a Venn diagram for this set.
In how many ways can \(\{1, . . . , n\}\) be written as the union of two sets? Here, for example, \(\{1, 2, 3, 4\}\cup\{4, 5\}\) and \(\{4, 5\}\cup\{1, 2, 3, 4\}\) count as the same way of writing \(\{1, 2, 3, 4, 5\}\) as a union.
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).
Between two mirrors \(AB\) and \(AC\), forming a sharp angle two points \(D\) and \(E\) are located. In what direction should one shine a ray of light from the point \(D\) in such a way that it would reflect off both mirrors and hit the point \(E\)?
If a ray of light comes towards a surface under a certain angle, it is reflected with the same angle as on the picture.
Consider a set of natural numbers \(A\), consisting of all numbers divisible by \(6\), let \(B\) be the set of all natural numbers divisible by \(8\), and \(C\) be the set of all natural numbers divisible by \(12\). Describe the sets \(A\cup B\), \(A\cup B\cup C\), \(A\cap B\cap C\), \(A-(B\cap C)\).
In good conditions, bacteria in a Petri cup spread quite fast, doubling every second. If there was initially one bacterium, then in \(32\) seconds the bacteria will cover the whole surface of the cup.
Now suppose that there are initially \(4\) bacteria. At what time will the bacteria cover the surface of the cup?
A piece containing exactly \(4\) black cells is cut out from a regular \(8\) by \(8\) chessboard. You are only allowed to cut along the edges of the cells and the piece must be connected - namely you cannot have cells attached only with a vertex, they have to share a common edge.
Find the largest possible area of such a piece.
Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.
In a distant village, there are \(3\) houses and \(3\) wells. Inhabitants of each house want to have access to all \(3\) wells. Is it possible to build non-intersecting straight paths from each house to each well? All houses and well must be level (that is, none of them are higher up, like on a mountain, nor are any of them on lower ground, like in a valley).